The mean of a set of five distinct numbers is 82. If the mean of the two lowest numbers is 65, and the mean of the two highest numbers is 100, what is the value of the median number when the five numbers are sorted in ascending order?

The correct answer is 85.

Step 1: Understand the relationships between total sum, mean, and count
The mean (average) of a set of numbers is calculated as:
$\text{Mean}=\frac{\text{Sum of numbers}}{\text{Total count of numbers}}$
By rearranging this formula, the sum of all numbers in the set is:
$\text{Sum of numbers}=\text{Mean}\times \text{Total count of numbers}$

Step 2: Calculate the total sum of all 5 numbers
We are given that the mean of the 5 distinct numbers is 82.
$\text{Total sum of 5 numbers}=5\times 82=410$

Step 3: Represent the sorted numbers and find partial sums
Let the five distinct numbers sorted in ascending order be represented as:
$x_1,x_2,x_3,x_4,x_5$
When 5 numbers are sorted in order, the middle number $x_3$ is the median.

1. The mean of the two lowest numbers ($x_1$ and $x_2$) is 65:
$\text{Sum of 2 lowest numbers}=x_1+x_2=2\times 65=130$

2. The mean of the two highest numbers ($x_4$ and $x_5$) is 100:
$\text{Sum of 2 highest numbers}=x_4+x_5=2\times 100=200$

Step 4: Solve for the median number
The total sum of all 5 numbers is equal to the sum of the two lowest numbers, plus the median number, plus the sum of the two highest numbers:
$(x_1+x_2)+x_3+(x_4+x_5)=410$
Substitute the values of the partial sums into the equation:
$130+x_3+200=410$
Combine the known constant values:
$330+x_3=410$
Subtract 330 from both sides to determine the median $x_3$:
$x_3=410-330=85$

Thus, the value of the median number is 85.